Equivalence of C^infinity atlases

Equivalence of $C^\infty$ atlases is an equivalence relation. Each $C^\infty$ atlas on $M$ is equivalent to a unique maximal $C^\infty$ atlas on $M$ .

Proof:

Reflexive: If $A$ is a $C^\infty$ atlas, then $A\cup A=A$ is also a $C^\infty$ atlas.

Symmetry: Let $A$ and $B$ be two $C^\infty$ atlases such that $A\cup B$ is also a $C^\infty$ atlas. Then certainly $B\cup A$ is also a $C^\infty$ atlas.

Transitivity: Let $A, B, C$ be $C^\infty$ atlases, such that $A\cup B$ and $B\cup C$ are both $C^\infty$ atlases.

Notation:
$\begin{aligned} A&=\{(U_\alpha,\varphi_\alpha)\}\\ B&=\{(V_\beta, \psi_\beta)\}\\ C&=\{(W_\gamma, f_\gamma)\}. \end{aligned}$

Then $\displaystyle \varphi_\alpha\circ f_\gamma^{-1}=\varphi_\alpha\circ\psi_\beta^{-1}\circ\psi_\beta\circ f_\gamma^{-1}: f_\gamma(U_\alpha\cap W_\gamma)\to\varphi_\alpha(U_\alpha\cap W_\gamma)$ is a diffeomorphism since both $\varphi_\alpha\circ\psi_\beta^{-1}$ and $\psi_\beta\circ f_\gamma^{-1}$ are diffeomorphisms due to $A\cup B$ and $B\cup C$ being $C^\infty$ atlases. Also, $M=\bigcup U_\alpha$ , $M=\bigcup W_\gamma$ implies $M=(\bigcup U_\alpha)\cup(\bigcup W_\gamma)$ so $A\cup C$ is also a $C^\infty$ atlas.

Let $A$ be a $C^\infty$ atlas on $M$ . Define $B$ to be the union of all $C^\infty$ atlases equivalent to $A$ . Then $B\sim A$ . If $B'\sim A$ , then $B'\subseteq B$ , so that $B$ is the unique maximal $C^\infty$ atlas equivalent to $A$ .